# Difference between revisions of "GenericAttacksMerkleDamgaard"

On this page, we describe attacks that apply on hash function that are designed according to the Merkle-Damgaard principle. Attacks that are even more generic and apply on all hash functions are described on this page

# 1 Collision style attacks

In case a hash function is not considered as a black box, but built from compression functions (which in turn are considered as black boxes at this point), multi-collisions can be constructed more efficiently. Ideally, the effort to find 2t single collisions should grow according to the birthday paradox: for t much smaller than n/2 the effort should grow almost linearly with each additional collision. What Joux showed in 2004 is that for iterated constructions the effort to find a 2t-multicollision is actually t*2n/2 The idea is to simply concatenate t collisions found by a birthday attack (or by any other mean like shortcut attacks for that matter). Since each collision allows to pick a message out of a pair of messages, and this choice is available t times, a set of 2t different messages consisting of t message blocks can be constructed that all lead to the same hash value.

An application of Joux's multicollisions is the analysis of concatenated constructions. Assuming two hash functions of output size n each whose outputs is concatenated, one would ideally expect a security of 2n against birthday based collision search attacks. Generating a 2n/2 multicollision for one of the hash functions is however enough to find a collision in the concatenated construction. This has a total cost of $2^{n/2+log(n)}$.

As a historic note, it should be mentioned that Coppersmith's attack in 1985 on the Davies-Price variant of Rabin's scheme builds already on such a multicollision idea.

Antoine Joux - Multicollisions in Iterated Hash Functions. Application to Cascaded Constructions

CRYPTO 3152:306-316,2004
Bibtex
Author : Antoine Joux
Title : Multicollisions in Iterated Hash Functions. Application to Cascaded Constructions
In : CRYPTO -
Date : 2004

Don Coppersmith - Another Birthday Attack

CRYPTO 218:14-17,1985
Bibtex
Author : Don Coppersmith
Title : Another Birthday Attack
In : CRYPTO -
Date : 1985

# 2 Second preimage attacks

Discoveries about second preimage attacks on iterated hash functions span more than two decades. Winternitz notes in 1984 that for messages of length $2^k$, the same number of different target hash values will speed-up the search for second preimages (of potentially different length) to $2^{n-k}$ trials.

Lai and Massey built up on that and also showed in 1992 that for first preimage larger than $2^{n/2}$, the effort to find a second preimage does not grow above $2 * 2^{n/2}$. Preneel further generalized this by taking storage requirements into account.

Robert S. Winternitz - A Secure One-Way Hash Function Built from DES

IEEE Symposium on Security and Privacy pp. 88-90,1984
http://doi.ieeecomputersociety.org/10.1109/SP.1984.10027
Bibtex
Author : Robert S. Winternitz
Title : A Secure One-Way Hash Function Built from DES
In : IEEE Symposium on Security and Privacy -
Date : 1984

Xuejia Lai, James L. Massey - Hash Function Based on Block Ciphers

EUROCRYPT 658:55-70,1993
Bibtex
Author : Xuejia Lai, James L. Massey
Title : Hash Function Based on Block Ciphers
In : EUROCRYPT -
Date : 1993

Bart Preneel - Analysis and Design of Cryptographic Hash Functions

Ph.D. Thesis, ,1999
Bibtex
Author : Bart Preneel
Title : Analysis and Design of Cryptographic Hash Functions
In : Ph.D. Thesis, -
Date : 1999

One of the reasons to include the message length as part of the message to be hashed in constructions since then, is to prevent these type of attacks. However, Dean describes in 1999 a way to circumvent this measure by used so-called expandable messages. Expandable messages are a set of messages of different lengths that all yield the same intermediate hash value.

Dean's construction only works for compression functions that have easily constructed fixed-points, i.e. where it is easy to find a message block and an input chaining value that results into the same output chaining value. Many popular hash function construction indeed do have this property. In 2005, Kelsey and Schneier managed to remove this constraint and gave an algorithm to construct expandable messages for any compression function with an n-bit intermediate value. Their idea is to construct multicollisions out of collisions between message blocks of different length. From that, again example messages can be constructed and hence the search for second preimages is again of order 2n-k+1word.

Very recently, Andreeva et al. extended this to cases in which there are multiple (say $2^t$) first preimages. Assuming a length of $2^k$ of each of them, it turns out that finding a single second preimage is equivalent to finding a second preimage for a message of $2^{k+t}$ message blocks. Also in this work it was shown that several constructions that employ dithering as a means to prevent previous generic second preimage fall to this new attack.

Richard D. Dean - Formal Aspects of Mobile Code Security

Ph.D. Thesis, ,1999
Bibtex
Author : Richard D. Dean
Title : Formal Aspects of Mobile Code Security
In : Ph.D. Thesis, -
Date : 1999

John Kelsey, Bruce Schneier - Second Preimages on n-Bit Hash Functions for Much Less than 2^n Work

EUROCRYPT 3494:474-490,2005
http://dx.doi.org/10.1007/11426639_28
Bibtex
Author : John Kelsey, Bruce Schneier
Title : Second Preimages on n-Bit Hash Functions for Much Less than 2^n Work
In : EUROCRYPT -
Date : 2005

Elena Andreeva, Charles Bouillaguet, Pierre-Alain Fouque, Jonathan J. Hoch, John Kelsey, Adi Shamir, Sebastien Zimmer - Second Preimage Attacks on Dithered Hash Functions

EUROCRYPT 4965:270-288,2008
Bibtex
Author : Elena Andreeva, Charles Bouillaguet, Pierre-Alain Fouque, Jonathan J. Hoch, John Kelsey, Adi Shamir, Sebastien Zimmer
Title : Second Preimage Attacks on Dithered Hash Functions
In : EUROCRYPT -
Date : 2008

# 3 Preimage attacks

Herding attacks are a special kind of preimage attack, in the sense that an additional assumption is being made for the attack to work. The basic scenario in which herding attacks are applicable is as follows. At the cost of a pre-computation step, an attacker can commit to a digest of a hash function without yet knowing the input. In the work of Kelsey and Kohno, this attack is described and shows that for all iterated hash functions the complexity is less than one would expect from an ideal hash function.

Resistance against herding attacks Given a hash function h, the attacker may choose a digest H. If she is given P, it should not be possible to find S such that h(P||S)=H is considerably faster than by 2n- invocations of h.

For short suffixes, the workfactor for a herding attack on an iterative hash functions is $2^{(2n-5)/3}$. First a so-called diamond structure is built in a precomputation phase that results in a particular digest H. After P is given to the attacker, a linking message $S_1$ is searched that connects P with one of the edges of the diamond structure. Let's denote the path between the found entry point in the diamond structure and the digest H at its end $S_2$, then the result string S such that h(P||S)=H is S=S_1 || S_2.

Besides observing this theoretical weakness, we can also consider the feasibility in practice of this attack. In the case of SHA-1, and without partial knowledge of P, a pre-computation effort of $2^{107}$ would be needed to compute H. This requires about $2^{60}$ bits of storage for the required data-structure. Afterwards, $2^{107}$ effort would be needed to compute S given a particular P, by search for a linking message block. This amounts to a total running time of $2^{108}$. If partial knowledge of P exists the attack can be much faster.

In order to exploit dedicated collision-search attacks on SHA-1, a collision search which is faster than about $2^{55.5}$ would be needed. Such a fast collision search would need to find a pair $(m,m^*)$ such that $h_c(cv_1,m)=h_c(cv_2,m^*)$ where the attacker has little control over the chaining variables $cv_1$ and $cv_2$. Such an algorithm is not known to date.

John Kelsey, Tadayoshi Kohno - Herding Hash Functions and the Nostradamus Attack

EUROCRYPT 4004:183-200,2006
http://dx.doi.org/10.1007/11761679_12
Bibtex
Author : John Kelsey, Tadayoshi Kohno
Title : Herding Hash Functions and the Nostradamus Attack
In : EUROCRYPT -